“Oppa, how do I look?”
Seoeun, wearing a cute pink dress, spun around once in front of Seoha.
Several days had passed since the event ended, but Seoeun was still completely enamored with the dress.
“You look incredibly pretty. It’s perfect for you, Seoeun!”
“Hehehe! Really? Can I wear this outside again?”
“Of course. Let’s go together next time too.”
“Then I’m going to practice walking prettily!”
Seoeun lightly held the hem of her dress like a princess and imitated a gait she seemed to have seen somewhere. Seoha smiled inwardly at how adorable she looked.
“Kids! Come eat!”
When they went downstairs, New Year’s tteokguk was ready on the table.
“Wow—tteokguk!”
Seoeun was the first to sit down.
The way she carefully arranged her skirt so the dress wouldn’t wrinkle made it seem as though she still hadn’t escaped from her game of make-believe.
“Let’s eat.”
So the children wouldn’t have to wait long, Cheolho quickly picked up his spoon.
Steaming hot tteokguk.
Thinly sliced rice cakes sat appetizingly in the milky-white ox bone broth. The egg garnish, green onions, and shredded seaweed sprinkled on top added color to the dish.
“Mmm. It’s delicious this year too.”
“Thank you, Mom.”
“Thank you for the meal!”
Miyeong nodded repeatedly with a proud expression.
This ordinary daily life, with the family gathered in one place sharing food, was more precious than anything to Miyeong. That was why, even though a publisher had requested her next essay collection, she never neglected preparing meals.
“Dad got promoted.”
Cheolho tossed out the words as if it were nothing special.
“Really?”
Seoha’s eyes widened.
An office position at a small and medium-sized shipping company affiliated with the school—that had been Cheolho’s first job in America. His main duties were checking inventory and serving as a liaison with Korea, but with his characteristic diligence and meticulousness, Cheolho fulfilled his role at 120 percent.
“It isn’t computerized?”
“Would someone who can do that be working in a place like this? The predecessor probably wrote things down in a notebook somewhere.”
The company’s management conditions were shocking.
‘I can’t stand this.’
Twenty years as a civil servant; he was a master of Excel.
In just one month, every inventory item in the warehouse had been computerized. It was work he had begun with neither barcodes nor a management system.
“We were losing this much?”
The management team leader muttered in disbelief as he looked at the Excel table.
But his work did not end there.
‘Don’t underestimate a Korean civil servant.’
He wrote functions so that ordering alerts would appear when inventory decreased, and overdue deliveries would be marked in red. Unit prices that had been managed by hand were soon changed to automatic calculations.
It did not take long for Cheolho to become a core member of the company.
“Applause for Dad!”
Clap, clap, clap!
At Miyeong’s words, Seoha and Seoeun stood up and applauded enthusiastically.
Cheolho shrugged.
He thought it was truly fortunate that, as the head of the household, he was now able to preserve at least the bare minimum of dignity.
* * *
Seoha had read thousands of papers on the Riemann Hypothesis.
Among them were quite a few studies that had been praised as groundbreaking breakthroughs. Analytic number theory, functional equations and symmetry, connections to the prime number theorem, random theory.
They had once been promising, but now all of them were known to have failed.
Why had the Riemann Hypothesis still not been solved despite the long efforts of so many mathematicians?
Was there a lack of mathematical techniques?
Had they been unable to keep up with the amount of computation?
Were they short on ideas?
Seoha shook his head.
It was none of those.
Analytic number theory had developed brilliantly for more than a century, and calculations of the zeros of the zeta function had expanded to a meaningless degree after the advent of quantum computers.
Approaches to the Riemann Hypothesis had also been so varied that there was almost nothing that had not been tried. But they shared one clear point in common. The moment they tried to seize the zeros of the Riemann zeta function, they quickly revealed their limits.
‘Why can the space of possible solutions not escape the region Riemann proposed?’
No one had been able to answer that question.
Seoha decided to pose a slightly more fundamental question.
‘Is the Riemann Hypothesis a question that asks whether something is true or false?’
This was not a philosophical question, but one of mathematical logic.
If it was true, why did every attempt stop at obtaining approximations?
If it was false, why had not a single counterexample disproving the hypothesis appeared?
All the numbers said that the Riemann Hypothesis was true, but it could not be proven.
“In cases like this, the direction is usually wrong.”
Many thoughts flashed through Seoha’s mind.
Gödel’s incompleteness theorem.
Within a complex axiomatic system, there exist propositions that are true but unprovable. And many logicians had already argued that the Riemann Hypothesis, too, might be one such problem.
Seoha shook his head.
‘I can’t go at it this way.’
This approach would ultimately only justify the claim that it could not be proven.
Seoha expanded his thinking to the realm of physics.
There was some kind of relationship between the Riemann Hypothesis and the movement of quanta. That was a fact he had already proven while observing quantum chaos.
‘The uncertainty principle?’
Just as position and momentum cannot be measured precisely at the same time, perhaps the existence of the zeros and the reason for them could not be captured simultaneously.
His mind tingled. His intuition was whispering that there was something related to the Riemann Hypothesis within the uncertainty principle.
Seoha put down his pencil and sat in his chair for a long while. One hour, then two passed, until the sun slowly began to set.
Then, suddenly, a word brushed through his mind.
‘Measurement?’
Mathematicians had always treated the zeros of the Riemann zeta function only as results.
Why do the zeros exist?
Where are they distributed?
But until now, no one had ever seriously asked:
‘What are we measuring?’
In quantum mechanics, a wave function has no physical quantity before observation.
It exists only as a structure of possibilities. And the moment observation occurs, that structure collapses and becomes fixed as a single value.
‘Riemann, just what did you discover?’
Natural numbers have no fixed position.
But the moment they pass through the zeta function, they gain period and phase.
The zeta function raised the discrete (離散, discrete—separate) objects known as natural numbers onto the continuous space known as the complex plane.
The irregularity of prime numbers was transformed into a series, and the series, through analytic continuation, was extended into a region where it had not even originally been defined.
‘All nontrivial zeros of the zeta function lie on the line with real part 1/2.’
No proof, no explanation.
Riemann had merely recorded a phenomenon plainly, as if writing in a diary.
To him, it might have been no more than a passing observation. Who could have known? That this one paper would torment countless mathematicians for such a long time.
Theo waved his hands as he looked at Seoha.
“There are more than a few insane mathematicians out there. Why else would even my old Stanford classmates contact me?”
After the presentation that day, Team Apex was receiving gazes mixed half with concern and half with anticipation.
Concern that Seoha might also collapse like the mathematicians who had fallen before the Riemann Hypothesis until now, and hope that perhaps he really might solve this devilish problem.
“But you’ve got a rough feel for it, haven’t you?”
Sujeong asked, looking at Seoha.
Seoha had always been like that. Even choices that seemed impulsive at first glance always had a reason.
“I had a question.”
“About the Riemann Hypothesis?”
“Yes. Mathematical thought usually has definitions and premises, and many flows follow from there. But it’s strange, isn’t it? Riemann’s case was completely different.”
“Now that you mention it…”
Sri stroked his chin.
“The observation came first. One day, the phenomenon just dropped out of the sky. Since it was so fascinating, mathematicians have been trailing after it, forcibly fitting structures onto it.”
“It is an extremely unusual case.”
They had been asking the reason and meaning behind the phenomenon, but they had not clearly defined what they had actually observed.
Like ancient people who must have been deeply shocked after witnessing lightning.
“So I made a conjecture. Perhaps there’s something that forces the zeros to appear at a specific position.”
The three of them were startled.
“Ah! So set the position aside for a moment?”
“Then that means the zeros are the product of necessity?”
Seoha nodded.
“If we assume that, we can neatly explain why no counterexamples have appeared until now. Of course, proof would also have been impossible unless we found out what that something was.”
Seoha walked over to the chalkboard and drew the complex plane.
“The zeta function transforms natural numbers into waves. Something is definitely happening in that process.”
“Like a Fourier transform?”
“Similar, but much more powerful.
The zeta function seems to impose an absolute constraint on integers. Just like how the string of a musical instrument can never produce sounds above a certain frequency.”
Seoha had also drawn inspiration from his conversation with Lawrence.
The zeta function ties natural numbers to a string. Then it fixes them to the frame called the real part of the complex plane.
Theo tried to work it through in his head. There was not a single contradiction in Seoha’s hypothesis.
“So the real part 1/2 is the maximum amplitude permitted by the zeta function… You wouldn’t be able to leave it unless you broke the system.”
Sri shouted as if he had just realized something.
“Ah! Then wouldn’t that also explain the relationship with quantum chaos that Seoha proved before?”
“The statistics of spacing between energy levels, random matrix theory.
People had already sensed it for decades. I merely proved that there was a rule to it.”
Seoha drew an energy level diagram beside the plane he had drawn on the chalkboard.
“This is only a conjecture, but the fact that the pattern appearing in quantum chaos and the distribution of the Riemann zeros coincide is not a coincidence.”
“Huh?”
“Ah… ah!”
“I see.”
The three of them reacted intensely.
“That’s right. If my hypothesis is correct, it would mean both are under the same kind of constraint.”
Was this evidence that they were drawing closer to the laws of the universe?
Physics and mathematics were connecting in a place no one had expected.
“I don’t think this is within the scope of what number theory can handle.”
Sujeong lowered her head with a gloomy expression.
Number theory dealt with static objects such as natural numbers, algebra, prime numbers, divisors, and congruence relations. But the path Seoha was now trying to take involved constraints and spectra.
It was fair to say there was nothing she could do there.
“No. It’s just beyond the scope of what current number theory can handle.”
“Hm?”
“This is a problem that absolutely must be handled through number theory.”
“Why?”
“Because the key is that integers and quanta are governed by the same law. I believe you’ll be able to find a way, noona.
If you solve it, number theory may have to be rewritten from its very foundations.”
Sujeong nodded with a serious expression.
“Just leave it to me.”
Rattle.
Sujeong opened the lab door and stepped into the hallway.
She had been walking normally, but the moment she turned the corner, she staggered dangerously.
She placed a hand against the wall and barely managed to keep her balance.
“Urgh!”
Nausea surged up.
The fluorescent lights in the hallway seemed to sway dizzily.
Sujeong hurriedly found the restroom and knelt in front of the toilet.
“Kuh-urk!”
Fortunately, she had eaten nothing all day.
Only gastric fluid came out.
Her hands trembled, and cold sweat ran over her entire body.
‘I believe you’ll be able to find a way, noona.’
What if I fail?
If I do, will you solve everything yourself like you used to?
“How long are you going to keep leaning on him? You came all this way to be of help to Seoha!”
Whoooosh—
‘Yes. I’m afraid.’
She admitted it.
She was afraid of disappointing Seoha. She was afraid she would hold the team back. She was terrified that she alone would fall behind before the most famous unsolved problem in the history of mathematics.
But what frightened her most was the possibility that Seoha would solve the problem without her help.
Sujeong clenched the edge beside the sink tightly. Then she muttered as if reminding herself.
“I can do this.”
Because Seoha had said she could.
Because his eyes had never once been wrong.